Routh Table Analysis for Pole Location

The problem asks to infer the location of system poles from the first two rows of a Routh table for a third-order characteristic equation. A standard third-order characteristic equation is given by $a_3s^3 + a_2s^2 + a_1s + a_0 = 0$. Based on the provided information, interpreted as coefficients related to the initial Routh table rows ($s^3$ coefficient assumed to be 1), we infer $a_3 = 1$, $a_2 = 33$, and $a_1 = 44$. The characteristic equation becomes $s^3 + 33s^2 + 44s + a_0 = 0$.

Routh Table Construction

The first two rows of the Routh table are constructed using the coefficients of the characteristic equation:

Calculating the $s^1$ Row

The elements of the $s^1$ row ($b_1, b_2$) are calculated using the Routh recurrence relation. The first element ($b_1$) is:

$b_1 = \frac{(33 \times 44) - (1 \times a_0)}{33} = 44 - \frac{a_0}{33}$

The second element ($b_2$) is calculated as 0.

Condition for Imaginary Axis Poles

Poles are located on the imaginary axis of the s-plane if a complete row of zeros appears in the Routh table. For this third-order system, this occurs when the first element of the $s^1$ row, $b_1$, is zero.

Determining the Value of $a_0$

Setting $b_1 = 0$ allows us to find the specific value of $a_0$ that leads to poles on the imaginary axis:

$44 - \frac{a_0}{33} = 0$

$\frac{a_0}{33} = 44$

$a_0 = 44 \times 33 = 1452$

Auxiliary Polynomial and Pole Location

When a row of zeros is encountered, the auxiliary polynomial is formed using the coefficients from the row immediately preceding it (the $s^2$ row). The auxiliary polynomial $A(s)$ is:

$A(s) = 33s^2 + a_0$

Substituting $a_0 = 1452$:

$33s^2 + 1452 = 0$

Solving for $s$ gives the locations of the poles:

$33s^2 = -1452$

$s^2 = -\frac{1452}{33} = -44$

$s = \pm \sqrt{-44} = \pm j\sqrt{44} = \pm j2\sqrt{11}$

These roots indicate a pair of complex conjugate poles situated exactly on the imaginary axis of the s-plane.